Strict monotonicity of principal eigenvalues of elliptic operators in $\mathbb{R}^d$ and risk-sensitive control.

This paper studies the eigenvalue problem on $\mathbb{R}^d$ for a class of second order, elliptic operators of the form $\mathcal{L}^f = a^{ij}\partial_{x_i}\partial_{x_j} + b^{i}\partial_{x_i} + f$, associated with non-degenerate diffusions. We present various results that relate the strict monotonicity of the principal eigenvalue of the operator with respect to the potential function $f$ to the ergodic properties of the corresponding `twisted' diffusion, and provide sufficient conditions for this monotonicity property to hold. Based on these characterizations, we extend or strengthen various results in the literature for a class of viscous Hamilton-Jacobi equations of ergodic type to equations with measurable drift and potential. In addition, we establish the strong duality for the equivalent infinite dimensional linear programming formulation of these ergodic control problems. We also apply these results to the study of the infinite horizon risk-sensitive control problem for diffusions. We establish existence of optimal Markov controls, verification of optimality results, and the continuity of the controlled principal eigenvalue with respect to stationary Markov controls.